Determining Optimal Level of Product Availability

Determining Optimal Level of Product Availability
SEG 4610 Supply Chain Management Determining Optimal Level of Product Availability Supply Chain Management Janny Leung

Learning Objectives Newsvendor model
Importance of level of product availability Factors to consider when setting availability levels Newsvendor model Managerial “levers” for improving supply chain profitability Value of postponement in a supply chain Setting optimal levels of product availability in practice Double marginalisation; contracts

Product Availability: Tradeoffs
High availability => responsive to customers attract increased sales higher revenue larger inventory higher costs risk of obsolescence Nordstrom, Marks & Spencer, Escada Bossini, supermarket, Fa Yuen Street E-commerce customer can find alternate source easily pressure on manufacturers to increase availability

Newsboy Model Newsvendor Model
single period model (one selling season) (one-time order, e.g. for quantity discount) demand uncertainty order placed (and delivered) before demand is known unmet demand is lost unsold inventory at the end of the period is discard (or salvaged at lower value) How much to order?

Factors affecting availability
Demand uncertainty Overstocking cost C0 = loss incurred when a unit unsold at end of selling season Understocking cost Cu = profit margin lost due to lost sale (because no inventory on hand) Customer/Cycle service level CSL =level of product availability = Prob(Demand < stock level)

Determining Optimal Level of Product Availability
Single period Possible scenarios Seasonal items with a single order in a season One-time orders in the presence of quantity discounts Continuously stocked items Demand during stockout is backlogged Demand during stockout is lost

Example: Selling parkas at LL Bean
Cost per parka = c = $45 Sale price per parka = p = $100 Inventory holding (until season end) and transportation cost (to outlet store) per parka = $10 Discount price per parka (season end sales) = $50 Salvage value per parka = $50 -$10 = $40 = s Cost of overstocking = Co = $45 + $10 - $50 = c - s = $ 5 Marginal profit from selling parka = cost of understocking = Cu = $100 - $45 = p - c = $55

L.L. Bean Example – Demand Distribution
Table 13-1 Demand Di (in hundreds) Probability pi Cumulative Probability of Demand Being Di or Less (Pi) Probability of Demand Being Greater than Di 4 0.01 0.99 5 0.02 0.03 0.97 6 0.04 0.07 0.93 7 0.08 0.15 0.85 8 0.09 0.24 0.76 9 0.11 0.35 0.65 10 0.16 0.51 0.49 11 0.20 0.71 0.29 12 0.82 0.18 13 0.10 0.92 14 0.96 15 0.98 16 17 1.00 0.00

LLBean: Expected Profit
Expected demand Expected profit if order 10 Expected profit if order k

LLBean: Expected profit

Newsvendor : Marginal Analysis
Stock one unit if … Stock 2 units (instead of 1 unit) if ... Stock 1 Stock 2 Stock 3 D = 0 D = 1 D = 2 D = 3

Increase order from k to k+1 if Prob(Demand < k) < Cu Co + Cu
Additional contribution keep order size at k 1 more unsold -Co Pk order k+1 instead of k Cu 1-Pk 1 fewer lost sale Order k+1 instead of k if Pr(D>k) Cu + Pr(D<k) (-Co) > 0 or Pr(D< k ) (-Co) + [1-Pr(D<k)] Cu > 0 Increase order from k to k+1 if Prob(Demand < k) < Cu Co + Cu

LL Bean

L.L. Bean Example Additional Hundreds Expected Marginal Benefit
Expected Marginal Cost Expected Marginal Contribution 11th 5,500 x 0.49 = 2,695 500 x 0.51 = 255 2,695 – 255 = 2,440 12th 5,500 x 0.29 = 1,595 500 x 0.71 = 355 1,595 – 355 = 1,240 13th 5,500 x 0.18 = 990 500 x 0.82 = 410 990 – 410 = 580 14th 5,500 x 0.08 = 440 500 x 0.92 = 460 440 – 460 = –20 15th 5,500 x 0.04 = 220 500 x 0.96 = 480 220 – 480 = –260 16th 5,500 x 0.02 = 110 500 x 0.98 = 490 110 – 490 = –380 17th 5,500 x 0.01 = 55 500 x 0.99 = 495 55 – 495 = –440 Table 13-2

L.L. Bean Example Figure 13-1

EXAMPLE A product is priced to sell at $100 per unit, and its cost is constant at $70 per unit. Each unsold unit has a salvage value of $30. Demand is expected to range between 35 and 40 units for the period: 35 units definitely can be sold and no units over 40 will be sold. The demand probabilities and the associated cumulative probability distribution (P) for this situation are shown on next slide. The marginal profit if a unit is sold is the selling price less the cost, or Cu = $100 − $70 = $30. The marginal loss incurred if the unit is not sold is the cost of the unit less the salvage value, or Co = $70 − $30 = $40. How many units should be ordered? SOLUTION The optimal probability of the last unit being sold is

According to the cumulative probability table (the last column in table below, 37 units should be stocked. The net benefit from stocking the 37th unit is the expected marginal profit minus the expected marginal loss. Demand and Cumulative Probabilities (p) CPn Number of Units Probability of Cumulative Demanded This Demand Probability to or more 1.00

Marginal Inventory Analysis for Units Having Salvage Value
(N) (p) (P) (MP) (ML) Units of Probability CPn Expected Marginal Expected Marginal Demand of Demand Profit of n-th Unit Loss of n-th Unit (Net) (100-70)(1- CPn-1) (70-30)CPn (MP)-(ML) $ $0 $30.00 (5.00) (22.50) (33.00) (40.00) Note: Expected marginal profit is the selling price of $100 less the unit cost of $70 times the probability the unit will be sold. Expected marginal loss is the unit cost of $70 less the salvage value of $30 times the probability the unit will not be sold. Net = (MP)(1 - CPn-1 ) - (ML) CPn-1 = ( )($100 - $70) - (0.75) ($70 - $30) = $ $10.00 = $12.50 For the sake of illustration, all possible decisions are shown. From the last column, we can confirm that the optimum decision is 37 units.

Newsvendor Model- Demand Distribution Continuous
SEG 4610 Supply Chain Management Newsvendor Model- Demand Distribution Continuous y Critical ratio Critical fractile Order y such that CSL* = Prob(Demand < y) = Cu Co + Cu Optimal Cycle Service level Janny Leung

Newsvendor model: normally distributed demand
Demand D ~ N(m,s) Order y such that CSL* = Prob(Demand < y*) = Cu Co + Cu Let y* = m+z*s

Optimal Cycle Service Level for Seasonal Items – Single Order
Co: Cost of overstocking by one unit, Co = c – s Cu: Cost of understocking by one unit, Cu = p – c CSL*: Optimal cycle service level O*: Corresponding optimal order size Expected benefit of purchasing extra unit = (1 – CSL*)(p – c) Expected cost of purchasing extra unit = CSL*(c – s) Expected marginal contribution of raising = (1 – CSL*)(p – c) – CSL*(c – s) order size

Optimal Cycle Service Level for Seasonal Items – Single Order

Evaluating the Optimal Service Level for Seasonal Items
Demand m = 350, s = 100, c = $100, p = $250, disposal value = $85, holding cost = $5 Salvage value = $85 – $5 = $80 Cost of understocking = Cu = p – c = $250 – $100 = $150 Cost of overstocking = Co = c – s = $100 – $80 = $20

Expected overstock Expected understock

Evaluating Expected Overstock and Understock
Expected understock

One-Time Orders in the Presence of Quantity Discounts
Using Co = c – s and Cu = p – c, evaluate the optimal cycle service level CSL* and order size O* without a discount Evaluate the expected profit from ordering O* Using Co = cd – s and Cu = p – cd, evaluate the optimal cycle service level CSL*d and order size O*d with a discount If O*d ≥ K, evaluate the expected profit from ordering O*d If O*d < K, evaluate the expected profit from ordering K units Order O* units if the profit in step 1 is higher If the profit in step 2 is higher, order O*d units if O*d ≥ K or K units if O*d < K

Evaluating Service Level with Quantity Discounts
Step 1, c = $50 Cost of understocking = Cu = p – c = $200 – $50 = $150 Cost of overstocking = Co = c – s = $50 – $0 = $50 Expected profit from ordering 177 units = $19,958

Evaluating Service Level with Quantity Discounts
Step 2, c = $45 Cost of understocking = Cu = p – c = $200 – $45 = $155 Cost of overstocking = Co = c – s = $45 – $0 = $45 Expected profit from ordering 200 units = $20,595

Desired Cycle Service Level for Continuously Stocked Items
Two extreme scenarios All demand that arises when the product is out of stock is backlogged and filled later, when inventories are replenished All demand arising when the product is out of stock is lost

Desired Cycle Service Level for Continuously Stocked Items
Q: Replenishment lot size S: Fixed cost associated with each order ROP: Reorder point D: Average demand per unit time σ: Standard deviation of demand per unit time ss: Safety inventory (ss = ROP – DL) CSL: Cycle service level C: Unit cost h: Holding cost as a fraction of product cost per unit time H: Cost of holding one unit for one unit of time. H = hC

Demand During Stockout is Backlogged
Increased cost per replenishment cycle of additional safety inventory of 1 unit = (Q > D)H Benefit per replenishment cycle of additional safety inventory of 1 unit = (1 – CSL)Cu

Lot size, Q = 400 gallons Reorder point, ROP = 300 gallons Average demand per year, D = 100 x 52 = 5,200 Standard deviation of demand per week, sD = 20 Unit cost, C = $3 Holding cost as a fraction of product cost per year, h = 0.2 Cost of holding one unit for one year, H = hC = $0.6 Lead time, L = 2 weeks Mean demand over lead time, DL = 200 gallons Standard deviation of demand over lead time, sL

Evaluating Optimal Service Level When Unmet Demand Is Lost
Lot size, Q = 400 gallons Average demand per year, D = 100 x 52 = 5,200 Cost of holding one unit for one year, H = $0.6 Cost of understocking, Cu = $2

Yield Management Airline, hotel bookings 2 classes of customers
high fare/revenue low fare/revenue Suppose there are infinite demand for low-fares Model: How many seats Q to allocate for high fares? C0 = LR Cu= HR - LR Overbooking?

Managerial levers for increased profitability
Increase salvage value Sell to outlet stores, overseas Decrease margin lost from stockout Backup sourcing (e.g. competitor?) Rain-check, discount coupon for future purchase Reduction of demand uncertainty Improve forecasting Quick response Postponement Tailored sourcing

Improved Forecasts Improved forecasts result in reduced uncertainty
Less uncertainty (lower sR) results in either: Lower levels of safety inventory (and costs) for the same level of product availability, or Higher product availability for the same level of safety inventory, or Both lower levels of safety inventory and higher levels of product availability An increase in forecast accuracy decreases both the overstocked and understocked quantity and increases a firm’s profits.

Impact of improved forecasts
Demand ~ N(350, sR), c=$100, p=$250, s=$85-$5= $80 Cost of understocking = Cu = p-c = $250-$100 = $150 Cost of overstocking = Co = c-s = $100 - $80 = $20 CSL* = Pr(D < y*) = ( )/(150+20)=0.88 y*= sR

Impact of Improved forecasts
y Expected profit y Expected overstock Expected understock s

Quick Response Reduction of replenishment leadtime
Allows for multiple orders during selling season Only if lead-time reduced sufficiently for additional orders to be executed before season ends Increased forecast accuracy Forecasts more accurate closer to selling season Forecast based on initial demand more accurate than pre-season forecasts Consequences of multiple replenishments: Expected total quantity less for same service level Average overstock (for disposal) is less Profits are higher

Quick Response: Multiple Orders Per Season
Ordering shawls at a department store Selling season = 14 weeks Cost per handbag = $40 Sale price = $150 Disposal price = $30 Holding cost = $2 per week Expected weekly demand = 20 SD of weekly demand = sD = 15

Two ordering policies Supply lead time is more than 15 weeks Single order placed at the beginning of the season Supply lead time is reduced to six weeks Two orders are placed for the season One for delivery at the beginning of the season One at the end of week 1 for delivery in week 8

Single Order Policy

Single Order Policy Expected profit with a single order = $29,767
Expected overstock = 79.8 Expected understock = 2.14 Cost of overstocking = $10 Cost of understocking = $110 Expected cost of overstocking = 79.8 x $10 = $798 Expected cost of understocking = 2.14 x $110 = $235

Two Order Policy Expected profit from seven weeks = $14,670
Expected overstock = 56.4 Expected understock = 1.51 Expected profit from season = $14, x $10 + $14,670 = $29,904

Impact of Quick Response

Three important consequences The expected total quantity ordered during the season with two orders is less than that with a single order for the same cycle service level The average overstock to be disposed of at the end of the sales season is less if a follow-up order is allowed after observing some sales The profits are higher when a follow-up order is allowed during the sales season

Figure 13-4

Figure 13-5

Two Order Policy with Improved Forecast Accuracy
Expected profit from second order = $15,254 Expected overstock = 11.3 Expected understock = 0.30 Expected profit from season = $14, x $10 + $15,254 = $30,488

Forecast Improves for Second Order (SD=3 Instead of 15)

Postponement Delay of product differentiation closer to time of sale.
Prior to point of postponement, only aggregate forecast needed (more accurate than individual product forecasts) Individual forecasts more accurate close to time of sale Better match of supply to demand, higher profits E.g. Benetton: dye  knit Valuable for on-line sales Costs?

Benetton Retail price p=$50, Salvage value s=$10
4 colours: demand for each ~ N(1000,5002) Option 1 (dye  knit): cost c=$20 Individual forecast 20 weeks ahead Option 2 (knit  dye): cost c=$22 Aggregate forecasts 20 weeks ahead Dye after individual demand known

Benetton Option 1: Option 2: CSL* = 30/40 = 0.75
y* = (0.674)500 =1337 Total production = 4(1337) = 5348 Expected overstock =1648, Expected understock =300 Expected profit = $94,576 Option 2: CSL* = 28/40 = 0.7 y* = (0.524)[2(500)] =4524 =total produced Expected overstock =715, Expected understock =190 Expected profit = $98,092

Value of Postponement: Benetton
Postponement is not very effective if a large fraction of demand comes from a single product Option 1 Red sweaters demand mred = 3,100, sred = 800 Other colors m = 300, s = 200 Expected profitsred = $82,831 Expected overstock = 659 Expected understock = 119

Other colors m = 300, s = 200 Expected profitsother = $6,458 Expected overstock = 165 Expected understock = 30 Total production = 3, x 435 = 4,945 Expected profit = $82, x $6,458 = $102,205 Expected overstock = x 165 = 1,154 Expected understock = x 30 = 209

Option 2 Total production = 4,475 Expected profit = $99,872 Expected overstock = 623 Expected understock = 166 Postponement may not be effective with Dominant Product

Value of Postponement Better match supply and demand
Increase profits, especially if firm produce large variety of products with similar demand level that is NOT positively correlated !! May reduce profits if there is major single product, especially if postponement increases manufacturing costs Tailored postponement Use postponement on uncertain demand Use lower-cost production on certain demand Segregate by product or by quantity

Tailored Postponement: Benetton
Use production with postponement to satisfy a part of demand, the rest without postponement Produce red sweaters without postponement, postpone all others Profit = $103,213 Tailored postponement allows a firm to increase profits by postponing differentiation only for products with uncertain demand

Separate all demand into base load and variation Base load manufactured without postponement Variation is postponed Four colors Demand mean  = 1,000,  = 500 Identify base load and variation for each color

Table 13-4 Manufacturing Policy Q1 Q2 Average Profit Average Overstock Average Understock 4,524 $97,847 510 210 1,337 $94,377 1,369 282 700 1,850 $102,730 308 168 800 1,550 $104,603 427 170 900 950 $101,326 607 266 1,050 $101,647 664 230 1,000 850 $100,312 815 195 $100,951 803 149 1,100 550 $99,180 1,026 211 650 $100,510 1,008 185

Tailored Sourcing Use a combination of two supply source:
One focused on lower cost, less able to handle uncertainty, One focused on flexibility but higher cost. Focus on different capabilities Better match supply to demand; increase profits Volume based: E.g. Benetton, firms with overseas suppliers Product based: E.g. Levi, traditional vs. custom jeans

Tailored Sourcing Strategies

Tailored Sourcing: Multiple Sourcing Sites

Dual Sourcing Strategies

Setting Product Availability for Multiple Products under Capacity Constraints
Single product order Multiple product order Decrease the order size Allocating the products When ordering multiple products under a limited supply capacity, the allocation of capacity to products should be based on their expected marginal contribution to profits. This approach allocates a relatively higher fraction of capacity to products that have a high margin relative to their cost of overstocking.

Setting Product Availability for Multiple Products Under Capacity Constraints
Two styles of sweaters from Italian supplier High end Mid-range m1 = 1,000 m2 = 2,000 s1 = 300 s2 = 400 p1 = $150 p2 = $100 c1 = $50 c2 = $40 s1 = $35 s2 = $25 CSL = 0.87 CSL = 0.80 O = 1,337 O = 2,337

Supplier capacity constraint, 3,000 units Expected marginal contribution high-end Expected marginal contribution mid-range

Set quantity Qi = 0 for all products i Compute the expected marginal contribution MCi(Qi) for each product i If positive, stop, otherwise, let j be the product with the highest expected marginal contribution and increase Qj by one unit If the total quantity is less than B, return to step 2, otherwise capacity constraint are met and quantities are optimal subject to:

Expected Marginal Contribution
Setting Product Availability for Multiple Products Under Capacity Constraints Expected Marginal Contribution Order Quantity Capacity Left High End Mid Range 3,000 99.95 60.00 2,900 99.84 100 2,100 57.51 900 2,000 800 57.00 1,300 780 54.59 920 300 42.50 43.00 1,000 1,700 200 36.86 1,800 180 39.44 1,020 40 31.89 30.63 1,070 1,890 30 30.41 1,080 10 29.67 29.54 1,085 1,905 1 29.23 29.10 1,088 1,911 29.09 1,089 Table 13-5

Setting Optimal Levels of Product Availability in Practice
Use an analytical framework to increase profits Beware of preset levels of availability Use approximate costs because profit-maximizing solutions are very robust Estimate a range for the cost of stocking out Ensure levels of product availability fit with the strategy

SEG 4610 Supply Chain Management
Summary Newsvendor model Tradeoff cost of over-stock and lost sales Managerial levers for increasing supply chain profitability Adjust costs Improve forecasting Quick response Postponement Tailored sourcing Allocate limited supply capacity among multiple products to maximise expected profits Life (cycle) is short! Making supply meet demand! Janny Leung

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